Hawking Radiation

Table Of Contents

General Relativity meets Quantum Mechanics

Steven Hawking studied quantum mechanics in curved space and found out that black holes emit thermal radiation with a temperature related to its size. The temperature of a black hole is directly related to the curvature of space-time, which is larger the smaller the black hole’s horizon is. Thus the smaller a black hole is the hotter it becomes. The hotter it is, the more intense radiation it emits. As the black hole loses mass due to it emitting thermal radiation, it shrinks, becomes even hotter, and emits even more radiation. This process leads to an exponential decrease of mass, eventually leading to an evaporation of the black hole in a final flash of light.

Black Holes evaporate

Black holes emit thermal radiation with a wavelength comparable to their diameter. Thus, smaller black holes shine hotter.

The images presented here depict the gravitational lens effect of a black hole which emits a thermal radiation. The smaller the black hole, the higher its temperature, according to Hawking radiation. The gravitational lens effect is caused by the deviation of light due to the immense gravity of the black hole and leads to a distortion of background stars. In this visualization, the Hawking radiation is strongly exaggerated: the actual radiation of an average black hole of the mass of the Sun would emit just a single photon in a long time scale, and it were of a wave length comparable to the size of the black hole (thus smaller black holes are warmer due to shorter wave lengths of the emitted light). Consequently a black hole would rather appear as a “fuzzy” object emitting a photon once per year (for a reasonable size) in an arbitrary direction. Only very small (i.e. at sub-microscopic scales) black holes emit radiation in an observable intensity, and then loose that much energy in such a short time such that they disappear soon after in a gamma ray flash.

Rendering Hawking Radiation

The following images were generated by volume rendering in curved space using raytracing, where at each volume element a thermal emission corresponding to the local curvature of space-time was assumed. This approach is simulating an atmosphere that gets hotter and brighter towards the event horizon. The light visible to the remote observer corresponds to the integral of the curved ray of light through this thermal atmosphere. It turns out that practically only the last steps close to the event horizon contribute, thus the black hole looks like a pretty homogeneous disk glowing in the temperature of the horizon, even though it is actually a volumetric object. As mentioned before, this approach is an unrealistic (though computationally very intense), artistic license to demonstrate some aspects of Hawking radiation.

A large black hole glows in a low temperature.

A smaller black hole comes with larger curvature of space-time, which corresponds to a higher radiation temperature.

Tiny black holes emit thermal radiation with high temperature, which in turn leads to higher loss of mass and, finally, evaporation.

The Final Universe

In the long term, all black holes evaporate. It takes a long amount of time for a black hole with a solar mass, but after 10¹⁰⁰ years even galactic black holes with (now) several million solar masses will have evaporated. This scenario depicts the final state of the universe, as we can foresee it by the current status of theoretical and observational physics: A vast empty space with a few photons left from the thermal radiation emitted by the last black holes.

The image sequence shown below is an artistic approach to visualize the final universe in 10¹⁰⁰ years, showing a black hole glowing in the foreground of other glowing black holes (the background objects are not stars, but black holes glowing in different colors according to their leftover masses), bending their background light (gravitational lensing), loosing mass due to its thermal radiation, shrinking, becoming hotter, and finally disappearing in a flash of intense light.

Animation Sequence

The animation of the evaporating black hole at the end of the universe is part of the video Tackling the Riddles of Gravity, starting at 3:30.

Trivia

The smallest possible black hole is one with a diameter of the Planck length. Such a black hole has a Planck mass, which is about 0.02mg. It is also known as the Planck particle and evaporates in 10^-39 seconds or ca. 90000 Planck times.

The size of a non-rotating, non-charged black hole with the mass of an electron – a black hole electron – has a diameter of 10^-57m, but when including rotation and electrical charge its radius is 2×10^-13m, which corresponds to the Compton wavelength of the electron. Such a black hole would be super-extremal and exhibit closed timelike curves.